A Lanczos-Stieltjes method for one-dimensional ridge function approximation and integration
For computational scientists performing uncertainty quantification or robust design, this method accelerates statistics computation by exploiting low-dimensional ridge structure, but it is an incremental improvement over existing ridge approximation techniques.
The paper develops a Lanczos-Stieltjes method for Gaussian quadrature and polynomial approximation of one-dimensional ridge functions, enabling efficient computation of output statistics by exploiting low-dimensional structure in high-dimensional input-output maps. The method approximates the univariate density via repeated convolutions and constructs orthogonal polynomials using Lanczos-Stieltjes, achieving accelerated uncertainty quantification and robust design.
Many of the input-parameter-to-output-quantity-of-interest maps that arise in computational science admit a surprising low-dimensional structure, where the outputs vary primarily along a handful of directions in the high-dimensional input space. This type of structure is well modeled by a ridge function, which is a composition of a low-dimensional linear transformation with a nonlinear function. If the goal is to compute statistics of the output (e.g., as in uncertainty quantification or robust design) then one should exploit this low-dimensional structure, when present, to accelerate computations. We develop Gaussian quadrature and the associated polynomial approximation for one-dimensional ridge functions. The key elements of our method are (i) approximating the univariate density of the given linear combination of inputs by repeated convolutions and (ii) a Lanczos-Stieltjes method for constructing orthogonal polynomials and Gaussian quadrature.